Hanson-Wright inequality and sub-gaussian concentration
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1 Electron. Commun. Probab. 18 (013, no. 8, 1 9. DOI: /ECP.v ISSN: X ELECTRONIC COMMUNICATIONS n PROBABILITY Hanson-Wrght nequalty and sub-gaussan concentraton Mark Rudelson Roman Vershynn Abstract In ths expostory note, we gve a modern proof of Hanson-Wrght nequalty for quadratc forms n sub-gaussan random varables. We deduce a useful concentraton nequalty for sub-gaussan random vectors. Two examples are gven to llustrate these results: a concentraton of dstances between random vectors and subspaces, and a bound on the norms of products of random and determnstc matrces. Keywords: subgaussan random varables; concentraton nequaltes; random matrces. AMS MSC 010: 60F10. Submtted to ECP on June 1, 013, fnal verson accepted on October 19, Hanson-Wrght nequalty Hanson-Wrght nequalty s a general concentraton result for quadratc forms n sub-gaussan random varables. A verson of ths theorem was frst proved n [9, 19], however wth one weak pont mentoned n Remark 1.. In ths artcle we gve a modern proof of Hanson-Wrght nequalty, whch automatcally fxes the orgnal weak pont. We then deduce a useful concentraton nequalty for sub-gaussan random vectors, and llustrate t wth two applcatons. Our arguments use standard tools of hgh-dmensonal probablty. The reader unfamlar wth them may beneft from consultng the tutoral [18]. Stll, we wll recall the basc notons where possble. A random varable ξ s called sub-gaussan f ts dstrbuton s domnated by that of a normal random varable. Ths can be expressed by requrng that E exp(ξ /K for some K > 0; the nfmum of such K s tradtonally called the sub-gaussan or ψ norm of ξ. Ths turns the set of subgaussan random varables nto the Orlcz space wth the Orlcz functon ψ (t = exp(t 1. A number of other equvalent defntons are used n the lterature. In partcular, ξ s sub-gaussan f an only f E ξ p = O(p p/ as p, so we can redefne the sub-gaussan norm of ξ as ξ ψ = sup p 1/ (E X p 1/p. p 1 One can show that ξ ψ defned ths way s wthn an absolute constant factor from the nfmum of K > 0 mentoned above, see [18, Secton 5..3]. One can smlarly defne sub-exponental random varables,.e. by requrng that ξ ψ1 = sup p 1 p 1 (E X p 1/p <. For an m n matrx A = (a j, recall that the operator norm of A s A = max x 1 Ax and the Hlbert-Schmdt (or Frobenus norm of A s A HS = (,j a,j 1/. Throughout the paper, C, C 1, c, c 1,... denote postve absolute constants. Partal support: M. R. by NSF grant DMS R. V. by NSF grant DMS and Unversty of Mchgan, U.S.A. E-mal: rudelson@umch.edu. Unversty of Mchgan, U.S.A. E-mal: romanv@umch.edu.
2 Theorem 1.1 (Hanson-Wrght nequalty. Let X = (X 1,..., X n R n be a random vector wth ndependent components X whch satsfy E X = 0 and X ψ K. Let A be an n n matrx. Then, for every t 0, P { X T AX E X T AX > t } [ ( t exp c mn K 4 A HS, t ] K. A Remark 1. (Related results. One of the ams of ths note s to gve a smple and selfcontaned proof of the Hanson Wrght nequalty usng only the standard toolkt of the large devaton theory. Several partal results and alternatve proofs are scattered n the lterature. Improvng upon an earler result on Hanson-Wrght [9], Wrght [19] establshed a slghtly weaker verson of Theorem 1.1. Instead of A = (a j, both papers had ( a j n the rght sde. The latter norm can be much larger than the norm of A, and t s often less easy to compute. Ths weak pont went unnotced n several later applcatons of Hanson-Wrght nequalty, however t was clear to experts that t could be fxed. A proof for the case where X 1,..., X n are ndependent symmetrc Bernoull random varables appears n the lecture notes of Nelson [14]. The moment nequalty whch essentally mples the result of [14] can be also found n [6]. A dfferent approach to Hanson-Wrght nequalty, due to Rauhut and Tropp, can be found n [8, Proposton 8.13]. It s presented for dagonal-free matrces (however ths assumpton can be removed by treatng the dagonal separately as s done below, and for ndependent symmetrc Bernoull random varables (but the proof can be extended to sub-gaussan random varables. An upper bound for P { X T AX E X T AX > t }, whch s equvalent to what appears n the Hanson Wrght nequalty, can be found n [10]. However, the assumptons n [10] are somewhat dfferent. On the one hand, t s assumed that the matrx A s postvesemdefnte, whle n our result A can be arbtrary. On the other hand, a weaker assumpton s placed on the random vector X = (X 1,..., X n. Instead of assumng that the coordnates of X are ndependent subgaussan random varables, t s assumed n [10] that the margnals of X are unformly subgaussan,.e., that sup y S n 1 X, y ψ K. The paper [3] contans an alternatve short proof of Hanson Wrght nequalty due to Latala for dagonal-free matrces. Lke n the proof below, Latala s argument uses decouplng of the order chaos. However, unlke the current paper, whch uses a smple decouplng argument of Bourgan [], hs proof uses a more general and more dffcult decouplng theorem for U-statstcs due to de la Peña and Montgomery-Smth [5]. For an extensve dscusson of modern decouplng methods see [4]. Large devaton nequaltes for polynomals of hgher degree, whch extend the Hanson-Wrght type nequaltes, have been obtaned by Latala [11] and recently by Adamczak and Wolff [1]. Proof of Theorem 1.1. By replacng X wth X/K we can assume wthout loss of generalty that K = 1. Let us frst estmate p := P { X T AX E X T AX > t }. Let A = (a j n,j=1. By ndependence and zero mean of X, we can represent X T AX E X T AX =,j = a j X X j a E X a (X E X + a j X X j.,j: j ECP 18 (013, paper 8. Page /9
3 The problem reduces to estmatng the dagonal and off-dagonal sums: { } p P a (X E X > t/ + P a j X X j > t/ =: p 1 + p.,j: j Step 1: dagonal sum. Note that X E X are ndependent mean-zero subexponental random varables, and X E X ψ1 X ψ1 4 X ψ 4. These standard bounds can be found n [18, Remark 5.18 and Lemma 5.14]. Then we can use a Bernsten-type nequalty (see [18, Proposton 5.16] and obtan p 1 [ ( t c mn, a t ] [ ( t exp c mn max a A, HS Step : decouplng. It remans to bound the off-dagonal sum S := a j X X j.,j: j t ]. (1.1 A The argument wll be based on estmatng the moment generatng functon of S by decouplng and reducton to normal random varables. Let λ > 0 be a parameter whose value we wll determne later. By Chebyshev s nequalty, we have p = P { S > t/ } = P { λs > λt/ } exp( λt/ E exp(λs. (1. Consder ndependent Bernoull random varables δ {0, 1} wth E δ = 1/. Snce E δ (1 δ j equals 1/4 for j and 0 for = j, we have S = 4 E δ S δ, where S δ =,j δ (1 δ j a j X X j. Here E δ denotes the expectaton wth respect to δ = (δ 1,..., δ n. Jensen s nequalty yelds E exp(λs E X,δ exp(4λs δ (1.3 where E X,δ denotes expectaton wth respect to both X and δ. ndces Λ δ = { [n] : δ = 1} and express S δ = a j X X j = ( X j a j X. Λ δ, j Λ c δ j Λ c δ Λ δ Consder the set of Now we condton on δ and (X Λδ. Then S δ s a lnear combnaton of mean-zero sub-gaussan random varables X j, j Λ c δ, wth fxed coeffcents. It follows that the condtonal dstrbuton of S δ s sub-gaussan, and ts sub-gaussan norm s bounded by the l -norm of the coeffcent vector (see e.g. n [18, Lemma 5.9]. Specfcally, S δ ψ Cσ δ where σδ := (. a j X j Λ c δ Λ δ Next, we use a standard estmate of the moment generatng functon of centered subgaussan random varables, see [18, Lemma 5.5]. It yelds E (Xj j Λ c δ exp(4λs δ exp(cλ S δ ψ exp(c λ σ δ. ECP 18 (013, paper 8. Page 3/9
4 Takng expectatons of both sdes wth respect to (X Λδ, we obtan E X exp(4λs δ E X exp(c λ σ δ =: E δ. (1.4 Recall that ths estmate holds for every fxed δ. It remans to estmate E δ. Step 3: reducton to normal random varables. Consder g = (g 1,..., g n where g are ndependent N(0, 1 random varables. The rotaton nvarance of normal dstrbuton mples that for each fxed δ and X, we have Z := ( g j a j X N(0, σδ. j Λ c δ Λ δ By the formula for the moment generatng functon of normal dstrbuton, we have E g exp(sz = exp(s σ δ /. Comparng ths wth the formula defnng E δ n (1.4, we fnd that the two expressons are somewhat smlar. Choosng s = C λ, we can match the two expressons as follows: E δ = E X,g exp(c 1 λz where C 1 = C. Rearrangng the terms, we can wrte Z = ( Λ δ X j Λ a c j g j. Then we can δ bound the moment generatng functon of Z n the same way we bounded the moment generatng functon of S δ n Step, only now relyng on the sub-gaussan propertes of X, Λ δ. We obtan E δ E g exp [C λ ( a j g j ]. Λ δ j Λ c δ To express ths more compactly, let P δ denotes the coordnate projecton (restrcton of R n onto R Λ δ, and defne the matrx A δ = P δ A(I P δ. Then what we obtaned ( E δ E g exp C λ A δ g. Recall that ths bound holds for each fxed δ. We have removed the orgnal random varables X from the problem, so t now becomes a problem about normal random varables g. Step 4: calculaton for normal random varables. By the rotaton nvarance of the dstrbuton of g, the random varable A δ g s dstrbuted dentcally wth s g where s denote the sngular values of A δ. Hence by ndependence, ( E δ = E g exp C λ s g = E g exp ( C λ s g. Note that each g has the ch-squared dstrbuton wth one degree of freedom, whose moment generatng functon s E exp(tg = (1 t 1/ for t < 1/. Therefore E δ ( 1 C λ s 1/ provded max C λ s < 1/. Usng the numerc nequalty (1 z 1/ e z whch s vald for all 0 z 1/, we can smplfy ths as follows: E δ exp(c 3 λ s = exp ( C 3 λ s provded max C 3 λ s < 1/. ECP 18 (013, paper 8. Page 4/9
5 Snce max s = A δ A and s = A δ HS A HS, we have proved the followng: E δ exp ( C 3 λ A HS for λ c0 / A. Ths s a unform bound for all δ. Now we take expectaton wth respect to δ. Recallng (1.3 and (1.4, we obtan the followng estmate on the moment generatng functon of S: E exp(λs E δ E δ exp ( C 3 λ A HS for λ c0 / A. Step 5: concluson. Puttng ths estmate nto the exponental Chebyshev s nequalty (1., we obtan Optmzng over λ, we conclude that p exp ( λt/ + C 3 λ A HS for λ c0 / A. [ ( t t ] p exp c mn A, =: p(a, t. HS A Now we combne wth a smlar estmate (1.1 for p 1 and obtan p = p 1 + p p(a, t. Repeatng the argument for A nstead of A, we get P { X T AX E X T AX < t } p(a, t. Combnng the two events, we obtan P { X T AX E X T AX > t } 4p(A, t. Fnally, one can reduce the factor 4 to by adjustng the constant c n p(a, t. The proof s complete. Sub-gaussan concentraton Hanson-Wrght nequalty has a useful consequence, a concentraton nequalty for random vectors wth ndependent sub-gaussan coordnates. Theorem.1 (Sub-gaussan concentraton. Let A be a fxed m n matrx. Consder a random vector X = (X 1,..., X n where X are ndependent random varables satsfyng E X = 0, E X = 1 and X ψ K. Then for any t 0, we have P { } ( AX A HS > t exp ct K 4 A. Remark.. The consequence of Theorem.1 can be alternatvely formulated as follows: the random varable Z = AX A HS s sub-gaussan, and Z ψ CK A. Remark.3. A few specal cases of Theorem.1 can be easly deduced from classcal concentraton nequaltes. For Gaussan random varables X, ths result s a standard consequence of Gaussan concentraton, see e.g. [13]. For bounded random varables X, t can be deduced n a smlar way from Talagrand s concentraton for convex Lpschtz functons [15], see [16, Theorem.1.13]. For more general random varables, one can fnd versons of Theorem.1 wth varyng degrees of generalty scattered n the lterature (e.g. the appendx of [7]. However, we were unable to fnd Theorem.1 n the exstng lterature. Proof. Let us apply Hanson-Wrght nequalty, Theorem 1.1, for the matrx Q = A T A. Snce X T QX = AX, we have E XT QX = A HS. Also, note that snce all X have unt varance, we have K 1/. Thus we obtan for any u 0 that P { AX A } [ HS > u exp C ( u K 4 mn A, u A T A HS ]. ECP 18 (013, paper 8. Page 5/9
6 Let ε 0 be arbtrary, and let us use ths estmate for u = ε A HS. Snce AT A HS A T A HS = A A HS, t follows that P { AX A } [ ] HS > ε A HS exp c mn(ε, ε A HS K 4 A. (.1 Now let δ 0 be arbtrary; we shall use ths nequalty for ε = max(δ, δ. Observe that the (lkely event AX A HS ε A HS mples the event AX A HS δ A HS. Ths can be seen by dvdng both sdes of the nequaltes by A HS and A HS respectvely, and usng the numerc bound max( z 1, z 1 z 1, whch s vald for all z 0. Usng ths observaton along wth the dentty mn(ε, ε = δ, we deduce from (.1 that P { } ( AX A HS > δ A HS exp cδ A HS K 4 A. Settng δ = t/ A HS, we obtan the desred nequalty..1 Small ball probabltes Usng a standard symmetrzaton argument, we can deduce from Theorem.1 some bounds on small ball probabltes. The followng result s due to Latala et al. [1, Theorem.5]. Corollary.4 (Small ball probabltes. Let A be a fxed m n matrx. Consder a random vector X = (X 1,..., X n where X are ndependent random varables satsfyng E X = 0, E X = 1 and X ψ K. Then for every y R m we have { P AX y < 1 } ( A HS exp c A HS K 4 A. Remark.5. Informally, Corollary.4 states that the small ball probablty decays exponentally n the stable rank r(a = A HS / A. Proof. Let X denote an ndependent copy of the random vector X. Denote p = P { AX y < 1 A HS}. Usng ndependence and trangle nequalty, we have { p = P AX y < 1 A HS, AX y < 1 } A HS P { A(X X < A HS }. (. The components of the random vector X X have mean zero, varances bounded below by and sub-gaussan norms bounded above by K. Thus we can apply Theorem.1 for 1 (X X and conclude that { P A(X X < } ( ( A HS t exp ct K 4 A, t 0. Usng ths wth t = (1 1/ A HS, we obtan the desred bound for (.. The followng consequence of Corollary.4 s even more nformatve. It states that AX y A HS + y wth hgh probablty. Corollary.6 (Small ball probabltes, mproved. Let A be a fxed m n matrx. Consder a random vector X = (X 1,..., X n where X are ndependent random varables K. Then for every y R m we have satsfyng E X = 0, E X = 1 and X ψ P { AX y < 1 6 ( A HS + y } ( exp c A HS K 4 A. ECP 18 (013, paper 8. Page 6/9
7 Proof. Denote h := A HS. Combnng the conclusons of Theorem.1 and Corollary.4, we obtan that wth probablty at least 1 4 exp( ch /K 4 A, the followng two estmates hold smultaneously: AX 3 h and AX y 1 h. (.3 Suppose ths event occurs. Then by trangle nequalty, AX y y AX y 3 h. Combnng ths wth the second nequalty n (.3, we obtan that ( 1 AX y max h, y 3 h 1 6 (h + y. The proof s complete. 3 Two applcatons Concentraton results lke Theorem.1 have many useful consequences. We nclude two applcatons n ths artcle; the reader wll certanly fnd more. The frst applcaton s a concentraton of dstance from a random vector to a fxed subspace. For random vectors wth bounded components, one can fnd a smlar result n [16, Corollary.1.19], where t was deduced from Talagrand s concentraton nequalty. Corollary 3.1 (Dstance between a random vector and a subspace. Let E be a subspace of R n of dmenson d. Consder a random vector X = (X 1,..., X n where X are ndependent random varables satsfyng E X = 0, E X = 1 and X ψ K. Then for any t 0, we have { d(x, } P E n d > t exp( ct /K 4. Proof. The concluson follows from Theorem.1 for A = P E, the orthogonal projecton onto E. Indeed, d(x, E = P E X, P E HS = dm(e = n d and P E = 1. Our second applcaton of Theorem.1 s for operator norms of random matrces. The result essentally states that an m n matrx BG obtaned as a product of a determnstc matrx B and a random matrx G wth ndependent sub-gaussan entres satsfes BG B HS + n B wth hgh probablty. For random matrces wth heavy-taled rather than sub-gaussan components, ths problem was studed n [17]. Theorem 3. (Norms of random matrces. Let B be a fxed m N matrx, and let G be an N n random matrx wth ndependent entres that satsfy E G j = 0, E G j = 1 and G j ψ K. Then for any s, t 1 we have P { BG > CK (s B HS + t n B } exp( s r t n. Here r = B HS / B s the stable rank of B. Proof. We need to bound BGx unformly for all x S n 1. Let us frst fx x S n 1. By concatenatng the rows of G, we can vew G as a long vector n R Nn. Consder the lnear operator T : l Nn l m defned as T (G = BGx, and let us apply Theorem.1 for T (G. To ths end, t s not dffcult to see that the the Hlbert-Schmdt norm of T equals B HS and the operator norm of T s at most B. (The latter follows from ECP 18 (013, paper 8. Page 7/9
8 BGx B G x B G HS, and from the fact the G HS s the Eucldean norm of G as a vector n l Nn. Then for any u 0, we have P { BGx > B HS + u } ( exp cu K 4 B. The last part of the proof s a standard coverng argument. Let N be an 1/-net of S n 1 n the Eucldean metrc. We can choose ths net so that N 5 n, see [18, Lemma 5.]. By a unon bound, wth probablty at least ( 5 n exp cu K 4 B, (3.1 every x N satsfes BGx B HS + u. On ths event, the approxmaton lemma (see [18, Lemma 5.] mples that every x S n 1 satsfes BGx ( B HS + u. It remans to choose u = CK (s B HS + t n B wth suffcently large absolutely constant C n order to make the probablty bound (3.1 smaller than exp( s r t n. Ths completes the proof. Remark 3.3. A couple of specal cases n Theorem 3. are worth mentonng. If B = P s a projecton n R N of rank r then P { P G > CK (s r + t n } exp( s r t n. The same holds f B = P s an r N matrx such that P P T = I r. In partcular, f B = I N we obtan { P G > CK (s N + t } n exp( s N t n. 3.1 Complexfcaton We formulated the results n Sectons and 3 for real matrces and real valued random varables. Usng a standard complexfcaton trck, one can easly obtan complex versons of these results. Let us show how to complexfy Theorem.1; the other applcatons follow from t. Suppose A s a complex matrx whle X s a real-valued random[ vector ] as before. Re A Then we can apply Theorem.1 for the real m n matrx à :=. Note that Im A ÃX = AX, à = A and à HS = A HS. Then the concluson of Theorem.1 follows for A. Suppose now that both A and X are complex. Let us assume that the components X have ndependent real and magnary parts, such that Re X = 0, E(Re X = 1, Re X ψ K, and smlarly [ for Im X ]. Then we can apply Theorem.1 for the real m n matrx Re A Im A A := and vector X Im A Re A = (Re X Im X R n. Note that A X = AX, A = A and A HS = A HS. Then the concluson of Theorem.1 follows for A. References [1] R. Adamczak, P. Wolff, Concentraton nequaltes for non-lpschtz functons wth bounded dervatves of hgher order, arxv: ECP 18 (013, paper 8. Page 8/9
9 [] J. Bourgan, Random ponts n sotropc convex sets. In: Convex geo- metrc analyss, Berkeley, CA, 1996, Math. Sc. Res. Inst. Publ., Vol. 34, 53 58, Cambrdge Unv. Press, Cambrdge (1999. MR [3] F. Barthe, E. Mlman, Transference Prncples for Log-Sobolev and Spectral-Gap wth Applcatons to Conservatve Spn Systems, arxv: [4] V. H. de la Peña, E. Gné, Decouplng. From dependence to ndependence. Randomly stopped processes. U-statstcs and processes. Martngales and beyond. Probablty and ts Applcatons (New York. Sprnger-Verlag, New York, MR [5] V. H. de la Peña, S. J. Montgomery-Smth, Decouplng nequaltes for the tal probabltes of multvarate U-statstcs, Ann. Probab. 3 (1995, no., MR [6] I. Dakonkolas, D. M. Kane, J. Nelson, Bounded ndependence fools degree- threshold functons, Proceedngs of the 51st Annual IEEE Symposum on Foundatons of Computer Scence (FOCS 010, Las Vegas, NV, October 3-6, 010. MR [7] L. Erdös, H.-T. Yau, J. Yn, Bulk unversalty for generalzed Wgner matrces, Probablty Theory and Related Felds 154, MR [8] A. Foucart, H. Rauhut, A Mathematcal Introducton to Compressve Sensng. Appled and Numercal Harmonc Analyss. Brkhäuser, 013. [9] D. L. Hanson, E. T. Wrght, A bound on tal probabltes for quadratc forms n ndependent random varables, Ann. Math. Statst. 4 (1971, MR [10] D. Hsu, S. Kakade, T. Zhang, A tal nequalty for quadratc forms of subgaussan random vectors, Electron. Commun. Probab. 17 (01, no. 5, 1 6. MR [11] R. Latala, Estmates of moments and tals of Gaussan chaoses, Ann. Probab. 34 (006, no. 6, MR [1] R. Latala, P. Mankewcz, K. Oleszkewcz, N. Tomczak-Jaegermann, Banach-Mazur dstances and projectons on random subgaussan polytopes, Dscrete Comput. Geom. 38 (007, MR-3114 [13] M. Ledoux, The concentraton of measure phenomenon. Mathematcal Surveys and Monographs, 89. Provdence: Amercan Mathematcal Socety, 005. MR [14] J. Nelson, Johnson Lndenstrauss notes, [15] M. Talagrand, Concentraton of measure and sopermetrc nequaltes n product spaces, IHES Publ. Math. No. 81 (1995, MR [16] T. Tao, Topcs n random matrx theory. Graduate Studes n Mathematcs, 13. Amercan Mathematcal Socety, Provdence, RI, 01. MR [17] R. Vershynn, Spectral norm of products of random and determnstc matrces, Probablty Theory and Related Felds 150 (011, MR [18] R. Vershynn, Introducton to the non-asymptotc analyss of random matrces. Compressed sensng, 10 68, Cambrdge Unv. Press, Cambrdge, 01. MR [19] E. T. Wrght, A bound on tal probabltes for quadratc forms n ndependent random varables whose dstrbutons are not necessarly symmetrc, Ann. Probablty 1 (1973, MR ECP 18 (013, paper 8. Page 9/9
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